[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29
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1 A p./29
2 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29
3 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) A p.2/29
4 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) f(x)dx f(x)dx = F(x) + C ( C ) A p.2/29
5 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) f(x)dx f(x)dx = F(x) + C ( C ) [ ] x α dx = α + xα+ + C (α ) A p.2/29
6 A p.3/29
7 [ ] x α dx = xα+ α + + C (α ) dx = log x + C x A p.4/29
8 [ ] x α dx = xα+ + C (α ) α + e x dx = e x + C dx = log x + C x A p.4/29
9 [ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = log x + C x cosxdx = sin x + C A p.4/29
10 [ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C A p.4/29
11 [ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x x 2 + dx = tan x + C dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C A p.4/29
12 [ ] x α dx = xα+ + C (α ) α + e x dx = e x + C sin xdx = cosx + C dx = tanx + C cos 2 x x 2 + dx = tan x + C x 2 dx = sin x + C dx = log x + C x cosxdx = sin x + C sin 2 x dx = tanx + C x 2 dx = cos x + C A p.4/29
13 A p.5/29
14 [ ] αf(x)dx = α f(x)dx ( α ) A p.6/29
15 [ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + g(x)dx A p.6/29
16 [ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + f(g(x))g (x)dx = F(g(x)) + C g(x)dx ( F(y) f(y) ) A p.6/29
17 [ ] αf(x)dx = α f(x)dx ( α ) {f(x) + g(x)} dx = f(x)dx + f(g(x))g (x)dx = F(g(x)) + C g(x)dx ( F(y) f(y) ) [ ] A p.6/29
18 A p.7/29
19 (ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + A p.8/29
20 (ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + (cosx) = sin x f(cosx) sin xdx = F(cosx)+C tan xdx = log cos x + C A p.8/29
21 (ax + b) = a F(x) f(x) f(ax + b)dx = f(ax + b)(ax + b) dx = F(ax + b)+c a a (ax + b) α dx = (ax + b)α+ + C (α ) a α + (cosx) = sin x f(cosx) sin xdx = F(cosx)+C tan xdx = log cos x + C ( x ) a + b = a ( x ) ( x ) ( x ) ( x ) f a +b dx = a f a +b a +b dx = af a +b +C x 2 + a 2dx = a 2 ( x 2 dx = a) + a tan x a + C A p.8/29
22 A p.9/29
23 [ ] (i) sin(ax + b)dx (a 0) (ii) (iii) (iv) (v) a x dx (a > 0) (a > 0) a2 x2dx f(tanx) x 2 a 2dx cos 2 x dx A p.0/29
24 [ ] (i) a cos(ax + b) + C A p./29
25 [ ] (i) a cos(ax + b) + C (ii) a x log a + C A p./29
26 [ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C A p./29
27 [ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C (iv) F(tanx) + C F (x) = f(x) A p./29
28 [ ] (i) a cos(ax + b) + C (ii) a x log a + C (iii) sin x a + C (iv) F(tanx) + C (v) 2a log x a x + a F (x) = f(x) + C (a > 0) A p./29
29 A p.2/29
30 [ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx A p.3/29
31 [ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) A p.3/29
32 [ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) [ ] log xdx = (x) log xdx A p.3/29
33 [ ] f (x)g(x)dx = f(x)g(x) f(x)g (x)dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) [ ] log xdx = (x) log xdx = x log x x x dx = x log x dx = x log x x + C A p.3/29
34 A p.4/29
35 [ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n A p.5/29
36 [ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] A dx (x 2 +A) n = (x 2 +A) x 2 (x 2 + A) n dx A p.5/29
37 [ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n dx = (x 2 +A) n n+ (x 2 +A) n ) x 2 dx A p.5/29
38 [ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n ) dx = (x 2 +A) x n n+ (x 2 +A) n 2 dx dx = (x 2 +A) x n 2( n+)(x 2 +A) + dx n 2( n+)(x 2 +A) n A p.5/29
39 [ ] n 2 (A > 0 ) dx (x 2 +A) = ( ) x 2n 3 dx + n A (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n [ ] dx (x 2 A (x 2 +A) = +A) x 2 n (x 2 + A) dx ( n ) dx = (x 2 +A) x n n+ (x 2 +A) n 2 dx dx = (x 2 +A) x n 2( n+)(x 2 +A) + dx n 2( n+)(x 2 +A) n x 3 dx = +2n (2n 2)(x 2 +A) n 2n 2 (x 2 +A) n A p.5/29
40 A p.6/29
41 [ ] ( ) ( ) A p.7/29
42 [ ] ( ) ( ) A p.7/29
43 [ ] ( ) ( ). A p.8/29
44 [ ] ( ) ( ). 2. (x a) n A p.9/29
45 [ ] ( ) ( ). 2. (x a) n 3. (x a) {(x a) 2 + b} n (b > 0) A p.20/29
46 [ ] ( ) ( ). 2. (x a) n (x a) {(x a) 2 + b} n (b > 0) {(x a) 2 + b} n (b > 0) A p.2/29
47 [ ] ( ) ( ). 2. (x a) n (x a) {(x a) 2 + b} n (b > 0) {(x a) 2 + b} n (b > 0) A p.22/29
48 A p.23/29
49 . A p.24/29
50 . 2. n + (n 2) log x a (x a) n A p.25/29
51 . 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b A p.26/29
52 . 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b 4. (x a) 2 + b dx = tan x a b b A p.27/29
53 . 2. n + (n 2) log x a (x a) n 3. 2( n + ) (n 2) {(x a) 2 + b} n 2 log (x a)2 + b 4. (x a) 2 + b dx = tan x a b b A p.28/29
54 38(75 ) 39(78 ) 42(83 46(92 ) A p.29/29
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3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
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y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t) =(x(t),y(t),z(t)) ( dp dx dt = dt, dy dt, dz ) dt f () > f x
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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
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t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
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I? 3 1 3 1.1?................................. 3 1.2?............................... 3 1.3!................................... 3 2 4 2.1........................................ 4 2.2.......................................
- 2 -
- 2 - - 3 - (1) (2) (3) (1) - 4 - ~ - 5 - (2) - 6 - (1) (1) - 7 - - 8 - (i) (ii) (iii) (ii) (iii) (ii) 10 - 9 - (3) - 10 - (3) - 11 - - 12 - (1) - 13 - - 14 - (2) - 15 - - 16 - (3) - 17 - - 18 - (4) -